IEPC-95-58 - 2 -

DIFFUSION MODEL OF THE CONDUCTION IN THE IONIZATION ZONEOF THE CLOSED ELECTRON DRIFT ACCELERATOR

V. 1. BARANOV. Yu. S. NAZARENKO, V. A. PETROSOV, A. 1. VASIN,Yu. M. YASHNOV*

Abstract

In this paper proposed is the diffusion model describing the current transfer in theclosed drift electron accelerator (CDA) from the acceleration layer to the anode. It is shown Uthan the intensive electron transfer crosswise the magnetic field in the ionization zone in aCDA channel may be satisfactorily explained on the basis of mechanisms of diffusion andheat conduction.

Developed is the related diffusion model for plasma motion in the ionization zone, onthe basis of this model the numerical calculations of a set of transfer differential equationsare performed, obtained are the distributions of the main plasma parameters: 3concentrations of atoms and charged particles, velocities of electrons and ions, electronstemperature. 3

Nomenclature

: - longitudinal coordinate;E. B - the longitudinal electric and radial magnetic field;j,, j - the densities of electrons and total longitudinal current;

a,_ - the conductivity crosswise a magnetic field;n, T - the concentration and temperature of electrons:

r., V - the concentration and velocity of atoms;rL - the Larmor radius of an electron;T - the time of electron collisions with atoms and ions;, - the ionization velocity of atoms;

E, - the ion price including the ionization potential and the radiation losses; I',. V - the velocities of ions and electrons along the channel;M - the ion mass;(P - the plasma potential;., - the nearwall potential in plasma;b - the channel width;

k= VJV1 - characterizes a ion flow to channel walls.

Introduction

As a rule, there is no difference between zones of ionization and acceleration in thelclosed electron drift accelerator (CDA). The abbreviation LIA (layer of ionization andacceleration) testifies to it. However, our experience reveals that in the majority of casesf(when considering conduction, oscillations, electron distribution function), the processes in5these zones proceed variously, therefore, these zones should be distinguished in theco!ntemporary CDA's iof T-100, T-160 type) [1, 2]. The most effective operation of aaIcctlerator is apparent:, ah.-ieved in the case, when :he ionization processes are mainl.comnplcied b\ the origi ;': :e accei .:4:i n zone.

Keds R arch ttte ofThe P ( -. Mosco. Russia.I' Kec'ivsn Research Ii.titute of Thermai Poces.'s (NilYPi. Moscow. Russia. .

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In the experiments, a great variety of versions of electric field (E) behavior in theionization zone [3] was observed, but the value of field remained enough small E<10 V/cm,with so regimes of accelerator operation in which the field changes its direction. Thissuggests that the electron conduction in the ionization zone must be defined by not classicvolume conduction j, = o E, but other mechanisms.

The "nearwall" conduction that is described in the enough large number of papers [4]is quite inadequate to explain an electron current in the ionization zone because of smallnessof a total electron flow reaching the walls what, in turn, is determined by a high nearwallpotential p, =6T [5]. The mechanism of electron conduction caused by oscillationsand instabilities in plasma is possible. But this problem needs separate and moredetailed consideration. It turns out that a small gradient of a charged particle concentration(-10 12 cm-/cm) is quite sufficient to explain the electron condition in the ionization zone.At the same time, it is clear that the gradient in the ionization zone is necessary to takeplace, since neutral atoms come out from anode, and all of them are practically ionized bythe beginning of the acceleration layer.

Set of equations for atoms and electrons

In the present paper, the diffusion mechanism for description of electrons transferfrom the acceleration layer to anode is proposed. The discharge model the basis of which isthe classic collision conduction in the E-LHfields is proposed in [5].

However, some inaccuracies are allowed for in it, which have resulted in conclusionsrather far from reality, for instance: electron pressure gradient was not tak,:n into account,the model was applied to the whole region of the acceleration channel whereas in the reeionU of the section (magnetic field maximum) the classic conduction is obviously inadequate toexplain an electron current.

Assuming that the mechanisms of diffusion conduction and heat conduction aredecisive in the ionization zone, one can write an appropriate set of equations for electronsand atoms. It incorporates continuity equations for atoms and electrons in terms ofionization and losses on the walls and electron energy equation. Instead of motionequations, used are the assumptions as to constancy of atoms velocities V=const anddiffusion model of transfer for electron.velocity V = -(r / )[(V(nT))] / nT.

Then. in one-dimensional approximation, a set has the form:

Vn' = -nn +(j- nV)k 2 / b;

S(nV) = np -(j- nV)k 2 / b (1)

5 (3/ 2)(nVT) = nnpjE, -(- nV)(k 2 / b)(, +[(- rf / r)(nT)].

Note that the choice of atoms and electrons but not atoms and ions, as the governingcomponents, is of fundamental importance. The motion of ions and atoms was consideredin [7], in doing so, the electric field was through as postulated (for example, fromexperimental measurements). But such an approach proves to be of small efficiency, sincewe obtain simply the accelerated motion of ions and the uniform motion of atoms in termsof ionization processes going in parallel. The motion of just electrons (not ions) isconsidered to be primary and governing in the ionization zone according to the mode!proposed in this paper, as the electrons are less mobile. than ions. crosswise ; magnetic field.

As electrons move to the anode under the effect of concentration gradient. tlheygenerate a current. At the same time, the charge separation generating an ,iectric field E(:)comes into e:xistence. This field adjusts automatically by itself to electron current j,(z), so

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that the total current \/ wili be constant 3t each s-ctnt (the case i7 point is the stito ;3:-regime without current oscillations in \olumL.i At su' aupproac . hen conside-.; the

motion exactl\ electrons, there is no necessit\ to introduce the field E(: as an add.:ion!parameter. It may be defined from the dependencies j,.[z, and ni). found already.

X (z) = - J /en;

m dV (2E(z) = -V --

e dz

Solution of a set of eauations and review of results

The proposed set of equations (1) for atoms and electrons in one-dimens;ona!statement reduces to Cauchy problem for a set of ordinary differential equations and allowsnumerical integration. It is more convenient to carry out the numerical integration alongelectron motion direction (to anode), then initial data are prescribed at the boundary ofionization and acceleration layers. When specifying the initial data, we proceed from thefollowing assumptions. The ion concentration and velocity at the channel exit are calculatedrather exactly by the known flow and thrust. Under the assumption that to the origin of theacceleration layer the whole mass flow involves practically ions which are already sped up 3-10eV, the ion concentration at the origin of the acceleration layer is easily found (from theflow equality). The initial temperature (energy) of electrons is determined by a drift velocityin the acceleration zone rather exactly. It is reasonable to specify the gradients ofconcentration and temperature as close to zero, as the concentration and temperature ofelectrons reach the peak value at the boundary of ionization and acceleration layers. Thechoice of the initial atom concentration is somewhat arbitrary. But in return, their 3concentration at the end of the layer, i.e. at the anode, is known exactly.

It is easily calculated by a mass flow under the assumption that the velocity of atomejection from the anode is determined by its temperature. Through variation of initial values 5of atom concentrations we select that value which. from the calculations, gives the value Uclosest to the true one near anode. Actual practice of calculations has revealed the efficiencyof such a method.

Obtained are the longitudinal distributions of the most important plasma parameters Iin the ionization zone: concentrations of atoms and charged particles, velocities of electronsand ions, temperature of electrons. The typical results of the performed calculations are Ishown in Fig. 1-3. Let us note the following main peculiarities for plasma parameterschange in the ionization zone that were obtained from solution of a set of equations (1).based on the diffusion model of electron conduction. -

The behavior of concentrations of neutral and charged particles is of certain interest. -Whereas ion concentration drops 1.5-2 times, atom concentration may increase tens times asterms of this fact one can use the atom balance equation for evaluation of ionization layerlength I

V, _ -n.n(T).dz a

The substitution of mean values for concentration n and ionization velocity P T) andintegration give:

no(z) = noo -exp nz = ne exp( .

Here z,o = Vo / np - the typical length of ionization where the neutral concentrationincreases "e" times.

I

U -42?'

N, 1VA*10- 11cm-

40

3 301 20

10

3Q 02 0.4 x, cm

I Fig. I

TI pe

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I I

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I t 1IIt is sh~o\n >t .ntens ive tren. .r Ie e>.o : se' .mon (tic2 7. H!

iO::zation zone of CDA ma> 'e expl:red hro:u i :: iNu merical s tior of ih set ( u?.Hr. b .< r dt. has n.:> m

:h receipt of onLu1n1.4. :str-o of nx:. .mc C cefveiocities and tempermre.

Apart from the fr:: tha:K et ot e:uanor e :c t-hi> ui-ier nd bcs.di orhe diffusion mechanis of elearons m:i'on ps:m u. OCCic. CiU:ation of satinrur

plasma parameters in tre ioniza:ion zon. it m as niti for research owlfrequenc\ ionization osci:atior emrerceMLe-

R efereric5

SA. S. Koroteex. V. A Pe:roso.. V. I BLrano sin. 3 R \We:c. S.-PDexelopmrent of -14 kW HJl-Tvne Ele:.ric Thrcster. I 1 -5

2. A. S. Korotecx. V. A Pe:roscc.. V. I. Bar.nor! A N r: R. \V:ci. S.- R \ WDe\elopment of a new Hi\\ l-E lent Lon-L if r -Tne El:::: Thrster of .1 7TV

Powe. i1-th Svnm-osi-. on Snac- Nuclear P L-: t- P-DPuis. .. Albuquerque. NeiMe\ico. Januarv 9-13. !9941.

3. A.I Buerova \. Krin-.. The Current St&e of Phivica Researches. in :neAcceleration Zone. In B : Plazmrienne uskoriceli r> i; nzektorx 1 "Nauka. 19>4.p. 107-129 (in Russian .

4. A. I. Buerova. A. I. Norozoi . Peculiari.;io of Ph\sic Processes in CDA. In Book:lonnve inzektor i plazmennve uskorteii. Moscov. Enereorrizdat. 1990. p.42- 5 inRussian).

5. V. 1. Barano%. A. I. Vasin. Yu. S. Nazarenke. V. A. Pc:.rosox. Yu. M. Yashn.Calculation of Near-Wall P&K:ential in a CDA-Channel. Letcers to ZiiTF, is in the press.1995. (in Russian).

6. V. K. Kalashnikov. Yu. V. Sanochkin. Discharge in the Accelerator With theClosed EIH Drift of Isothermal Electrons. Fizika pazmv. \. '. issue 2. p. 303-311, 19S1 (inRussian).

7. B. 1. Volkox, A. G. Sveshnikov. S. A. Yakunin. On Mathematical Simuiation ofPhysical Processes in Plasma-Optics Ssstems. DokI. AN USSR. 197S, \. 23. p. 26-268 (inRussian).

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